On Complex Lie Algebras with a Simple Real Form

Seminar Sophus Lie 1 (1991) 243–245

On complex Lie algebras with a simple real form

S. S. L. Darmstadt

The following observations, motivated by a question of Lawson’s [1], does not seem to be readily accessible in the literature. Let a denote a complex Lie algebra and g a real subalgebra. Then g+ig and g ig are complex subalgebras. If X g then [X, g ig] [X, g] i[X, g] g ig.∩ Then [iX, g ig] = i[X, g ig] ∈i(g ig) = g ∩ig. ⊆Thus g∩ ig is an⊆ ideal∩ of g + ig. ∩ ∩ ⊆ ∩ ∩ ∩

Remark 1. If g is simple, then either g = g ig, i.e., ig = g and g is a ∩ complex subalgebra, or else g ig = 0 , i.e. g + ig = g ig ∼= C g is the complexiﬁcation of g embedded∩ into a{. } ⊕ ⊗

Lemma 2. Assume g1 ig1 = C g1 and g2 ig2 = C g2 , and suppose that ⊕ ∼ ⊗ ⊕ ∼ ⊗ there is a morphism f: g1 g2 of real Lie algebras. Then F : g1 ig1 g2 ig2 , F (X + iY ) = f(X) + if(Y→) is a morphism of complex algebras.⊕ → ⊕

Proof. Compute or observe that under the isomorphisms gj + igj = C gj ∼ ⊗ the map F corresponds to the map id f: C g1 C g2 ! C ⊗ ⊗ → ⊗ If a is the complexiﬁcation of a real simple Lie algebra g which itself is the underlying real vector space of a complex Lie algebra then a = g ig has a real involution τ given by τ(X + iY ) = X iY which is a complex⊕conjugate automorphism of a, satisfying τ(iZ) = iτ−(Z) for all Z a. The complex − + ∈ Lie algebra a is isomorphic to the direct sum g g− of two simple complex +⊕ ideals which are conjugate complex (i.e., g− = τ(g )). These may nevertheless + be isomorphic as complex algebras. Each real isomorphism f: g g− gives a + + → subalgebra gf = X f(X): X g which is isomorphic to g = g. We call { ⊕ ∈ } + ∼ these graph subalgebras of a. If we deﬁne σ: g g− by σ(X) = τ(X), i.e., by → restricting and corestricting τ , then g = gσ . The graph subalgebra gf is complex + if and only if f: g g− is a complex isomorphism. A real automorphism of a → will be called special if it leaves the ideals g invariant.

Lemma 3. For two graph subalgebras gf and gg there is a special real automorphism F of a with F (gf ) = gg . If f = σ and gg is complex, then F is neither complex nor complex conjugate. 1 Proof. We note that gf − : g− g− is an automorphism α of g− , and g = α f . We deﬁne a special automorphism→ F : a a by F (X Y ) = X α(Y ). Now w◦e have g = X g(X): X g+ = →X α(f(X⊕)): X ⊕g+ = g { ⊕ ∈ } { ⊕ ∈ } 244 Darmstadt

F X f(X): X g = F (gf ). Suppose that f = σ and that g is complex, { ⊕ + ∈ } then F g = idg+ is complex and F g− = α is complex conjugate. Thus F is neither |complex nor complex conjugate.|

Proposition 4. Let a be a complex Lie algebra with a simple real form g and a real subalgebra g∗ which is isomorphic to g. Then one of the following mutually exclusive cases occurs:

(A) There is a complex automorphism F : a a with F (g) = g∗ . → (B) a is not simple, and there is a special real autmorphism F : a a which → is neither complex nor complex conjugate with F (g) = g∗ . + (C) a is not simple, and g∗ is one of the two unique ideals g or g− + with a = g g− . There is no isomorphism of a mapping g to g∗ , ⊕ but the projection onto g∗ is a complex endomorphism of a mapping g isomorphically onto g∗ .

Proof. Since g∗ ∼= g, the real subalgebra g∗ is simple. By Remark 1, we have two cases: Case (i): g∗ ig∗ . Case (ii): ig∗ = g∗ . ⊕ In Case (i), Lemma 2 shows the existence of an endomorphism F of a + with F (g) = g∗ . If a is simple, then F must be an automorphism. If a = g g− then the complex endomorphism F must respect the two summands. If it ⊕were zero on one of the summands, then g would have to be contained in the other, but g is contained in neither. Hence F is an automorphism and we are in Case (A). In Case (ii), because of g ∼= g∗ , the algebra a is the complexiﬁcation of the underlying real algebra of a complex simple Lie algebra and is therefore of + the form g = g g− according to remarks preceding Lemma 3. The following ⊕ + mutually exclusive cases occur: (a) g∗ = g . (b) g∗ = g− . (c) g∗ is a complex graph subalgebra. In Cases (a) and (b) we have Case (C) of the Proposition. Then g∗ is an ideal while g is not. Hence no real or complex automorphism of a maps g onto g∗ . In Case (c) we apply Lemma 3 to the two graph subalgebras g = gσ and g∗ and ﬁnd ourselves in Case (B) of the Proposition.

Corollary 5. If a is a simple complex Lie algebra and g a real form, and if g∗ is a real subalgebra of a which is isomorphic to g then there is a complex automorphism of a mapping g onto g∗ .

We remark, that the automorphism group Aut(g) of a complex simple Lie algebra is a semidirect product of the normal connected subgroup Int(g) = ead X : X g of inner automorphisms by a ﬁnite group E , namely, the auto- h ∈ i morphism group of a basis of a root system. The group E is of order 2 for An , n > 1, Dn , n = 4, E6 and of order 3 for D4 , and it is trivial in all other cases. (See [3] or [2], 6 notably p. 298.)

Corollary 6. Let A be a complex simple and simply connected Lie group with complex conjugation κ, and let G denote the real form of elements fixed under κ. Let H denote any real analytic subgroup of A whose Lie algebra is isomorphic Darmstadt 245 to g = L(G). Then H is conjugate to G under a complex automorphism of A. In particular, A is closed and isomorphic to H . Proof. The Lie algebra a of A is the complexification of g and is simple. Hence we are in Case (A) of Proposition 4 and find a complex automorphism F : a a with F (g) = h. We have Aut g = (Inn g) E = Ad(G) E with a finite→group E or order 1,2, or 3 according to the list×above. Since ×A is simply connected, there is a unique automorphism f: A A inducing F . The group H is connected (see e.g. [2], p. 214, Theorem 9).→The analytic groups H and f(G) have the same Lie algebra and are both connected. Hence they agree. If the group of automorphisms of a fixing g as a whole contains an automorphism which is not inner, or if there are no outer automorphisms of a then the group of inner automorphisms acts transitively on the set of subalgebras isomorphic to g. The algebra sl(nC) (realizing type An 1 ) for instance admits − a non-inner involution of the form X X > which leaves sl(n, R) invariant. A 7→ − similar statement holds for Dn , n odd, and E6 (see [2], 297f.) In such cases, in Corollary 6, the groups G and H are conjugate under an inner automorphism.

References [1] Lawson, J. D., A question on the simple real forms of the Lie algebra sp(2,C), e-mail message, October 1991. [2] Onishchik, A. L., and E. B. Vinberg, “Lie Groups and Algebraic Groups”, Springer-Verlag Berlin etc., 1990. [3] Tits, J., “Tabellen zu den einfachen Lie Gruppen und ihren Darstellun- gen”, (Springer) Lecture Notes in Mathematics 40 (1967).

Karl H. Hofmann Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstraße 7 D-6100 Darmstadt, Germany

Received October 10, 1991